Breast anatomy modeling using Neutro-Connectedness

Breast anatomy represents the structure of the breast and is useful for breast tumor detection and classification in clinical practice. Breast contains four primary layers: skin layer, fat layer, mammary layer, and muscle layer [47]. BUS image regions in different layers have different appearances (e.g, intensities and textures); and breast tumor mainly exists in the mammary layer (see Fig. 4.4). Breast anatomy was utilized in [2,26] for tumor segmentation. In [2], Shao identified two horizontal lines to remove the fat and muscle by applying phase congruency [46] and Otsu’s thresholding. However, it was difficult to identify the two horizontal lines accurately; in some cases, part of the tumor could be divided into the fat region. In this work, we propose a new Neutro-Connectedness (NC) [29] based framework that models breast anatomy by incorporating region similarity and image depth. It decomposes BUS images into 3 to 5 layers.

There are two components in NC: the degree of connectedness t and confidence of connectedness c, NC( , ) = [ ( , ), ( , )] where i and j indicate the ith and jth pixel or region, respectively. Image regions from the same layer have strong connectedness (e,g., high t and c values), and from different layers have weak connections. NC builds on the following three fundamental concepts:

(1) NC of two adjacent regions i and j. The degree of connectedness of two adjacent regions is defined as their similarity, noted as ; and the degree of confidence is defined as the homogeneity between them, noted as .

(2) NC of a path. The degree of connectedness of a path is defined as the minimum value of along the path, and degree of confidence is the minimum value along the path.

(3) NC of any two regions. The degree of connectedness is defined by the strongest path between the two regions. It uses the confidence of the corresponding path as the degree of connectedness confidence of the two regions.

NC computation generates NC maps that demonstrate the degrees of connectedness and confidence among image regions, and NC trees that show regions’ structure. Each NC tree contains a group of image regions that share common properties. All NC trees form an natural decomposition of an image. In this work, NC is applied to decompose BUS images into different layers; and we redefine the NC of two adjacent regions by utilizing the region similarity and image depth. The depth term penalizes the growth of NC trees along the vertical direction to avoid the cross-layer expanding.

( , , ) = ( , ) ∙ ( , ) (36)

( , ) = (ℎ( ), ℎ( )) (37) ( , ) = (−| ( ) − ( )|/ ) (38)

( , ) = (−| ( ) − ( )|/ ) (39) In Eq. (36), denotes the similarity between the ith and jth regions, and is the normalized depth difference between the ith region and the root region (k) of a NC tree. In Eq. (37), ℎ(∙) defines the homogeneity of a region [29, 30]. In Eq. (38), ( ) and ( ) are the normalized intensities of the ith and jth regions, respectively; ( ) denotes the row index of the ith region center. and control the shapes of the two exponential functions. is 0.5 the same as [44]. Without the depth term in Eq. (39), the layer of the region i will be determined by the connectedness values between region i and different root regions. For illustration, the ith region is at the bottom of the image, and the root region k is at the top of the image, and the jth layer is between them. If the connectedness between

the ith region and the root region k is the maximum among the connectednesses of the ith region with the root regions, and the ith region would be expanded into the layer of the root region k. In such a case, the NC tree expanded by the root region k would cross the group

j, which does not match the breast anatomy. The expanding of the layer along the vertical

direction should be avoided, but only along the horizontal direction using Eq. (39). controls the span of the layer, and larger will result in fewer layers. The initial is chosen by applying values from 0.1 to 0.5 with a step size of 0.1 on the training dataset (refer section 4.3.2) to generate the layers. Eighty percent of the training dataset images were decomposed into three to five layers when = 0.2, and the other images were decomposed into 6 or more layers. However, we can control the layer number to be 3 to 5 by initializing as 0.2 and updating it adaptively. If the number of layers is greater than five, decrease by 0.05; and increase by 0.05, otherwise.

After computing the NC of two adjacent regions, the connectednesses of a path and Fig. 4.5. Effectiveness of different . (a) = 0.2, before merging;(b)- (e): with = 0.05, 0.1, 0.15 and 0.2, after merging, respectively. From top to bottom, the same color indicates the same layer.

between any two regions can be calculated easily. The left boundary regions of an image are set as the roots for generating NC trees. All the regions on a tree are in a group (layer). If a layer cannot cover more than 75% of the image width, it will be merged into its nearest layer. The effectiveness of the merging step with different is shown in Fig. 4.5 (a) and (e). Note that each generated image layer is composed of a group of image regions that have high connectedness with each other; those regions have high probability from the same biological tissue layer, but the generated image layer is not the biologic tissue layer.